Ikeda¶

Systems / Maps / Chaotic maps

The Ikeda map — a spiral strange attractor from a nonlinear optical ring cavity.

discrete · map2 dimensionschaotic

Definition¶

@staticmethod
def _step(X, a, b, u):
    x, y = X
    t = a - b / (1 + x**2 + y**2)
    xp = 1 + u * (x * np.cos(t) - y * np.sin(t))
    yp = u * (x * np.sin(t) + y * np.cos(t))
    return xp, yp

Parameters¶

Symbol	Default	Role
`a`	`0.4`	phase-offset constant of the cavity detuning
`b`	`6`	nonlinear phase-modulation strength
`u`	`0.9`	round-trip field-loss (dissipation) factor

Properties¶

Lyapunov spectrum

$+0.501,\; -0.7117$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.704$

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

3 fixed points
1 stable · 2 unstable

Define it in TSDynamics¶

import tsdynamics as ts

sys = ts.systems.Ikeda()
traj = sys.iterate(steps=10_000)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference¶

Ikeda (1979), Opt. Commun. 30, 257-261

BibTeX

@misc{ikeda,
  title = {Ikeda system},
  note = {Ikeda (1979), Opt. Commun. 30, 257-261}
}